Fig. 10

Boolean backward equivalence shown on a simple example: not all attractors are preserved. (Top-left) BN with three variables denoted by \(x_1\), \(x_2\), and \(x_3\). (Bottom-left) The underlying STG. The model has 4 steady-state attractors (nodes 100, 010, 000, and 111). Two have same activation values for \(x_1\) and \(x_2\) (000, and 111), two have not. (Top-right) Variables \(x_1\) and \(x_2\) can be shown to be BBE-equivalent by inspecting their update functions. If they have the same value in a state, i.e. \(x_1(t)=x_2(t)\), then they will be equivalent for all successor states. Based on this, a reduced BN can be obtained by considering a representative variable for each block and rewriting the corresponding update functions in terms of those representatives (here the representative variable is denoted by \(x_{1,2}\)). (Bottom-right) The underlying STG agrees with the original one on all states that have equal values for variables in the same block (purple nodes in bottom-left panel). Notably, the two attractors having same activation value for \(x_1\) and \(x_2\) are preserved, while the other two are dropped, as expected by our theory